Remark 1.1 In our theorem REF , if \(\theta =0\) , then it is the Hölder continuous Euler flow with compact support and have been constructed by P.Isett and Sung-jin Oh in [1]}. In fact, they construct H\(\ddot{o}\) lder continuous Euler flow with H\(\ddot{o}\) lder exponent \(\frac{1}{5}-\varepsilon \) . Moreover, if we take \(a(t,x)=10\varepsilon \)  in \(Q_r\)  and \(N\)  sufficiently large, we must have \(\theta \ne 0.\)

